They sound like the same idea and aren't. Object permanence is Piaget's: a hidden ball still exists — first signs around eight to twelve months, full understanding nearer two years, and no settled scientific consensus on the timing. Object constancy is Mahler's: a person you're furious with is still the person you love, which takes years and, for some people, never fully settles.
Side by side
Aspect
Object constancy
Object permanence
Theorist
Mahler
Piaget
About
People
Things
Domain
Emotional
Cognitive
Arrives
Years
~8–24 months
What each one means
Object Constancy
In psychoanalysis, the capacity to keep a stable, whole picture of someone — and stay connected to your good feelings about them — when they're absent, disappointing, or you're angry with them. In perceptual psychology the same phrase means something else entirely: seeing an object's size, shape and colour as constant even as distance, angle and light change.